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This module extends the ideas of the Discrete Fourier Transform (DFT) into two-dimensions, which is necessary for any image processing.

2d dft

To perform image restoration (and many other useful image processing algorithms) in a computer, we need a FourierTransform (FT) that is discrete and two-dimensional.

F k l u 2 k N v 2 l N F u v
for k 0 N 1 and l 0 N 1 .
F u v m n f m n u m v m
F k l m N 1 0 n N 1 0 f m n 2 k m N 2 l n N
where the above equation ( ) has finite support for an N x N image.

Inverse 2d dft

As with our regular fourier transforms, the 2D DFT also has an inverse transform that allows us to reconstruct an imageas a weighted combination of complex sinusoidal basis functions.

f m n 1 N 2 k N 1 0 l N 1 0 F k l 2 k m N 2 l n N

Periodic extensions

Illustrate the periodic extension of images.
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2d dft and convolution

The regular 2D convolution equation is

g m n k N 1 0 l N 1 0 f k l h m k n l

Below we go through the steps of convolving two two-dimensional arrays. You can think of f as representing an image and h represents a PSF, where h m n 0 for m n 1 and m n 0 . h h 0 0 h 0 1 h 1 0 h 1 1 f f 0 0 f 0 N 1 f N 1 0 f N 1 N 1 Step 1 (Flip h ):

h m n h 1 1 h 1 0 0 h 0 1 h 0 0 0 0 0 0
Step 2 (Convolve):
g 0 0 h 0 0 f 0 0
We use the standard 2D convolution equation ( ) to find the first element of our convolved image. In order to better understand what ishappening, we can think of this visually. The basic idea is to take h m n and place it "on top" of f k l , so that just the bottom-right element, h 0 0 of h m n overlaps with the top-left element, f 0 0 , of f k l . Then, to get the next element of our convolved image, we slide the flipped matrix, h m n , over one element to the right and get the following result: g 0 1 h 0 0 f 0 1 h 0 1 f 0 0 We continue in this fashion to find all of the elements ofour convolved image, g m n . Using the above method we define the general formula to find a particular element of g m n as:
g m n h 0 0 f m n h 0 1 f m n 1 h 1 0 f m 1 n h 1 1 f m 1 n 1

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Circular convolution

What does H k l F k l produce?

2D Circular Convolution

g ~ m n IDFT H k l F k l circularconvolutionin2D

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Due to periodic extension by DFT ( ):

Based on the above solution, we will let

g ~ m n IDFT H k l F k l
Using this equation, we can calculate the value for each position on our final image, g ~ m n . For example, due to the periodic extension of the images, when circular convolution is applied we willobserve a wrap-around effect.
g ~ 0 0 h 0 0 f 0 0 h 1 0 f N 1 0 h 0 1 f 0 N 1 h 1 1 f N 1 N 1
Where the last three terms in are a result of the wrap-around effect caused by the presence of the images copies located all around it.

Zero padding

If the support of h is M x M and f is N x N , then we zero pad f and h to M N 1 x M N 1 (see ).

Circular Convolution = Regular Convolution

Computing the 2d dft

F k l m N 1 0 n N 1 0 f m n 2 k m N 2 l n N
where in the above equation, n N 1 0 f m n 2 l n N is simply a 1D DFT over n . This means that we will take the 1D FFT of each row; if wehave N rows, then it will require N N operations per row. We can rewrite this as
F k l m N 1 0 f m l 2 k m N
where now we take the 1D FFT of each column, which means that if we have N columns, then it requires N N operations per column.
Therefore the overall complexity of a 2D FFT is O N 2 N where N 2 equals the number of pixels in the image.

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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Source:  OpenStax, Intro to digital signal processing. OpenStax CNX. Jan 22, 2004 Download for free at http://cnx.org/content/col10203/1.4
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